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Ch. 64:Structural Econometric Modeling4283f (y x), the conditional density of y given x;E(y x), the conditional expectation of y given x;Cov(y x), the conditional covariances (or correlations) of y given x; or,Qα (y x) the α conditional quantile of y given x.The most commonly estimated characteristic of the joint density is the best linear predictor (BLP(y x)) of y given x.More recently researchers have taken advantage of developments in nonparametric and semiparametric statistical methods to derive consistent estimates of the jointdensity of y and x. For example, statisticians have proposed kernel density techniquesand other data smoothing methods for estimating f (x, y). These same smoothing techniques have been used to develop nonparametric conditional mean models. Silverman(1986), Härdle (1990), Härdle and Linton (1994) and others provide useful introductions to these procedures. A major advantage of nonparametric models is that they canprovide flexible descriptions of the above statistical quantities.Given their flexibility, it would seem that nonstructural empirical researchers shouldalways prefer nonparametric methods. There are, however, limitations to nonparametricmethods. One is that they may require large amounts of data to yield much precision.1Second, and more important, once estimated, it is unclear how a flexible estimate of ajoint density can be used to recover economic constructs such as economies of scale inproduction and consumer welfare. Moreover, it is also unclear how to perform out-ofsample counterfactual calculations, such as the impact of an additional bidder on thewinning bid in an auction.It is tempting to look at our descriptions of structural versus nonstructural models,and parametric versus nonparametric models, and see them as absolutes – empiricalmodels are either structural or nonstructural, parametric or nonparametric. We see littlevalue in such absolute classification exercises. In practice, it is not uncommon to findstructural econometric models that include nonstructural components or nonparametric components. Our goal in providing these definitions is to have an initial basis forclassifying and evaluating the success of an econometric model.An initial example from IO may help understand our focus and intent. Consider aresearcher who observes the winning bids, y {y1 , . . . , yT } , in each of a large number of T similar auctions. Suppose the researcher also observes the number of bidders,x {x1 , . . . , xT } , in each auction. To understand the equilibrium relationship betweenwinning bids and the number of bidders the researcher could use a structural or a nonstructural approach.1 Silverman (1986) argues that researchers using these techniques face a “curse of dimensionality”, whereinthe amount of data required to obtain precise estimates grows rapidly with the dimensions of x and y. Hiscalculations (1986, Table 4.2) suggest that researchers may need thousands, if not hundreds of thousands ofobservations before they can place great faith in these flexible techniques. For instance, more than ten timesas much data is required to attain the same level of precision for a four-dimensional as a two-dimensionaljoint density. More than 200 times as much data is required for an eight-dimensional as a four-dimensionaldensity.

4284P.C. Reiss and F.A. WolakA standard nonstructural approach would be to regress winning bids on the numberof bidders. Under standard statistical assumptions, this regression would deliver the bestlinear predictor of winning bids given the number of bidders. These coefficient estimatescould be used to predict future winning bids as a function of the number of bidders. Alternatively, a researcher worried about a nonlinear relationship between winning bidsand the number of bidders might instead opt to use nonparametric smoothing techniques to estimate the conditional density of winning bids given each distinct observednumber of bidders x, f (y x). The researcher could then use this estimated conditionaldensity, f (y x), to calculate whether, for example, expected winning bids in the sample auctions increased or decreased with the number of bidders. The researcher couldalso check to see if the conditional expected bid increased or decreased linearly withthe number of bidders.The process of formulating and implementing either of these nonstructural modelsso far has made little use of economics, except perhaps to identify what is y and whatis x. For instance, our discussion of these descriptive models has made no referenceto institutional features of the auctions (e.g., sealed-bid versus open-outcry and firstprice versus second-price). It also has not required economic assumptions about bidderbehavior or characteristics (e.g., risk-neutrality, expected profit maximization and bidder competition). In some cases (e.g., sealed-bid versus open-outcry), we may be ableto incorporate these considerations into a nonstructural model by introducing them asconditioning variables. In other cases (e.g., the degree of risk aversion), this may not bepossible.A key reason then to use economic theory, beyond specifying x and y, is to clarifyhow institutional and economic conditions affect relationships between y and x. Thisspecificity is essential once the researcher wishes to make causal statements about estimated relationships or use them to perform counterfactuals. Suppose for example, theresearcher has regressed winning bids on the number of bidders and estimates the coefficient on the number of bidders is 100. Can this estimate be interpreted as the causaleffect of adding another bidder to a future auction? We would argue that without furtherknowledge about the institutional and economic features of the auctions under study theanswer is no. What separates structural models from nonstructural models, and somestructural models from others, is how clearly the connections are made between institutional, economic, and statistical assumptions and the estimated relationships. While it ispossible to assert that assumptions exist that make the estimated relationship causal, theplausibility of such claims ultimately rests on whether these assumptions are reasonablefor the researcher’s application.As we discuss later in Section 8, IO economists have developed a variety of structuralmodels of auction bid data. These models have been used to derive causal models of theequilibrium relations between winning bids and the number of bidders. Paarsch (1992),for example, was the first to compare empirical models of winning bids in private valueand common value sealed-bid auctions. For instance, he showed that for sealed-bidauctions with risk-neutral, expected profit-maximizing, Pareto-distributed-private-valuebidders would have the following density of winning bids y given a known number of

Ch. 64:Structural Econometric Modeling4285bidders, x:f (y x, θ ) θ2 xy θ2 x 1 θ1 θ2 (x 1)θ2 (x 1) 1 θ2 x.Using this density, Paarsch derives the expected value of the winning bid conditional onthe number of bidders: θ2 xθ1 θ2 (x 1)E(y x) .θ2 (x 1) 1 θ2 x 1Paarsch’s paper motivated IO economists to think about what types of institutional,economic and statistical assumptions were necessary to recover causal relationshipsfrom auction data. For example, researchers have asked how risk aversion, collusionand asymmetric information change the equilibrium distribution of bids. Researchersalso have compared the observable implications of using sealed-bid versus open-outcryauctions.In closing this section, we should re-emphasize the general goal of structural econometric modeling. Structural econometric models use economic and statistical assumptions to identify economic quantities from the joint density of economic data, f (x, y).The main strength of this approach is that, done right, it can make clear what economicassumptions are required to draw causal economic inferences from the distribution ofeconomic data.3. Constructing structural modelsHaving introduced the concept of a structural model, we now explore how structuralmodelers go about constructing econometric models.3.1. Sources of structureThere are two general sources of “structure” in structural models – economics and statistics. Economics allows the researcher to infer how economic behavior and institutionsaffect relationships between a set of economic conditions x and outcomes y. Often theseeconomic models are deterministic, and as such do not speak directly to the distributionof noisy economic data. Structural econometric modelers thus must add statistical structure in order to rationalize why economic theory does not perfectly explain data. As weshow later, this second source of structure may affect which economic quantities a researcher can recover and which estimation methods are preferable.In any structural modeling effort a critical issue will be: How did the structuralmodeler know what choices to make when introducing economic and statistical assumptions? Most answers to this question fall into one of three categories: those madeto reflect economic realities; those made to rationalize what is observed in the data or describe how the data were generated; and, those made to simplify estimation. We should

4286P.C. Reiss and F.A. Wolaknote at the outset that there is no necessary agreement among structural modelers as tohow to make these choices. Some purists, for example, believe that structural modelsmust come from fully-specified stochastic economic models. Others find it acceptableto add structure if that structure facilitates estimation or allows the researcher to recover economically meaningful parameters. For instance, economic theory may makepredictions about the conditional density of y given x, f (y x), but may be silent aboutthe marginal density of x, f (x). In this case, a researcher might assume that the marginal density of x does not contain parameters that appear in the conditional density. Ofcourse, there is nothing to guarantee that assumptions made to facilitate estimation arereasonable.The “structure” in a structural model is there because the researcher explicitly orimplicitly chose to put it there. Although we have argued that one of the advantagesof a structural econometric model is that researchers can examine the sensitivity of thestructural model estimates to alternative assumptions, this is sometimes easier said thandone.The following example illustrates how some of these issues can arise even in a familiar linear regression setting. Specifically, we ask what types of assumptions are requiredto interpret a regression of outputs on inputs as a production function.E XAMPLE 1. Imagine an economist with cross-section, firm-level data on output, Qt ,labor inputs, Lt , and capital inputs, Kt . To describe the relationship between firm i’soutput and inputs, the economist might estimate the regression:ln Qt θ0 θ1 ln Lt θ2 ln Kt t ,(1)by ordinary least squares (OLS). In this regression, the θ ’s are unknown coefficients andthe t is an error term that accounts for the fact that the right-hand side input variablesdo not perfectly predict log output.What do we learn by estimating this regression? Absent more information we haveestimated a descriptive regression. More precisely, we have estimated the parameters ofthe best linear predictor of yt ln(Qt ) given xt (1, ln(Lt ), ln(Kt )) for our sampleof data. Goldberger (1991, Chapter 5) provides an excellent discussion of best linearpredictors. The best linear predictor of y given a univariate x is BLP(y x) a bx,where a E(y) bE(x) and b Cov(y, x)/ Var(x). Absent more structure, thecoefficients a and b are simply functions of population moments of f (x, y).If we add to our descriptive model the assumption that the sample second momentsconverge to their population counterpartsT1 xt xt MxxT Tlimt 1andT1 xt yt Mxy ,T Tlimt 1and that Mxx is a matrix of full rank, then OLS will deliver consistent estimates of theparameters of the best linear predictor function. Thus, if we are interested in predictingthe logarithm of output, we do not need to impose any economic structure and very little

Ch. 64:Structural Econometric Modeling4287statistical structure to estimate consistently the linear function of the logarithm of laborand logarithm of capital that best predicts (in a minimum-mean-squared-error sense) thelogarithm of output.Many economists, however, see regression (1) as more than a descriptive regression.They base their reasoning on the observation that (1) essentially looks like a logarithmicβrestatement of a Cobb–Douglas production function: Qt A Lαt Kt exp( t ). Becauseof the close resemblance, they might interpret (1) as a production function.A critical missing step in this logic is that a Cobb–Douglas production function typically is a deterministic relationship for the producer, whereas the regression model (1)includes an error term. Where did the error term in the empirical model come from? Theanswer to this question is critical because it affects whether OLS will deliver consistentestimates of the parameters of the Cobb–Douglas production function, as opposed toconsistent estimates of the parameters of the best linear predictor of the logarithm ofoutput given the logarithms of the two inputs. In other words, it is the combination ofan economic assumption (production is truly Cobb–Douglas) and statistical assumptions ( t satisfies certain moment conditions) that distinguishes a structural model froma descriptive one.Deterministic production function models provide no guidance about the propertiesof the disturbance in (1). The researcher thus is left to sort out what properties are appropriate from the details of the application. One could imagine, for instance, the modelerdeclaring that the error is an independently-distributed, mean-zero measurement errorin output, and that these errors are distributed independently of the firms’ input choices.In this case, OLS has the potential to deliver consistent estimates of the productionfunction parameters.But how did the modeler know that t was all measurement error? As we discusslater this is likely too strong an assumption. A more plausible assumption is that theerror also includes an unobservable (to the econometrician) difference in each firm’sproductivity (e.g., an unobservable component of At in the Cobb–Douglas function).The existence of such components raises the possibility that the input choices are correlated with t . Such correlations invalidate the use of OLS to recover consistent estimatesof the parameters of the Cobb–Douglas production function.Even if one were willing to assume that t is measurement error distributed independently of xt , additional economic structure is necessary to interpret the OLS parameterestimates as coefficients of a Cobb–Douglas production function. By definition, a production function gives the maximum technologically feasible amount of output that canbe produced from a vector of inputs. Consequently, under this stochastic structure, unless the researcher is also willing to assert that the firms in the sample are producingalong their Cobb–Douglas production function, OLS applied to (1) does not yield consistent estimates of the parameters of this production function. In a theoretical realm, theassumption of technologically efficient production is relatively innocuous. However, itmay not fit the institutional realities of many markets. For example, a state-owned firmmay use labor in a technologically inefficient manner to maximize its political capitalwith unions. Regulators also may force firms to operate off their production functions.

4288P.C. Reiss and F.A. WolakBoth of these examples underscore the point that care must exercised to ensure that theeconomic model fits the institutional reality of what is being studied.This example demonstrates what assumptions are necessary for a linear regressionmodel to have a causal economic interpretation as a production function. First, theresearcher must specify an economic model of the phenomenon under consideration,including in this case the functional form for the production function. Second, she mustincorporate unobservables into the economic model. This second step should receivesignificantly more attention than it typically does. This is because the assumptions madeabout the unobservables will impact the consistency of OLS parameter estimates. Sections 4 and 5 further illustrate the importance of stochastic specifications and potentialpitfalls.3.2. Why add structure?We see three general reasons for specifying and estimating a structural econometricmodel.First, a structural model can be used to estimate unobserved economic or behavioralparameters that could not otherwise be inferred from nonexperimental data. Examplesof structural parameters include: marginal cost; returns to scale; the price elasticity ofdemand; and the impact of a change in an exogenous variable on the amount demandedor on the amount supplied.Second, structural models can be used to perform counterfactuals or policy simulations. In counterfactuals, the researcher uses the estimated structural model to predictwhat would happen if elements of the economic environment change. For example,suppose that we have estimated the demand for a product and the monopolist’s costfunction. We could then, with added assumptions, use these estimates to calculatehow market prices and quantities would change if an identical second firm entered themonopoly market.For these calculations to be convincing, the structural modeler must be able to argue that the structural model will be invariant to the contemplated change in economicenvironment. Thus, if we were to evaluate instead the effect of a regulator capping a monopolist’s price, we would have to maintain that the monopolist’s cost function wouldnot change as a result of the regulation. That these assumptions need to be made andchecked is again another illustration of the value of a structural model – it can help researchers identify what assumptions are required in order to draw inferences and makepredictions about economic behavior and phenomena.Finally, structural models can be used to compare the predictive performance of twocompeting theories. For example, we could compare the performance of quantity-settingversus price-setting models of competition. It is important to emphasize that these comparisons do not provide unambiguous tests of the underlying economic theories. Indeed,these comparisons are always predicated on untestable assumptions that are not partof the theory. For instance, any “test” of quantity-setting behavior versus price-setting

Ch. 64:Structural Econometric Modeling4289behavior is predicated on the maintained functional forms for demand, costs, and theunobservables. Thus, the only sense in which one can “test” the two theories is to askwhether one of these ways of combining the same economic and stochastic primitivesprovides a markedly better description of observed or out-of-sample data.Because we cannot test economic models independent of functional form assumptions for a finite number of observations, it is important to recognize that structuralparameter estimates may well be sensitive to these assumptions. For example, if wewere trying to estimate consumer surplus, we should be aware that it might make atremendous difference that we assumed demand was linear, as opposed to constantelasticity. While this sensitivity to functional form can be viewed as a weakness, italso can be viewed a

5.2.3. The stochastic model 4320 5.3. Summary 4324 6. Market power models more generally 4325 6.1. Estimating price–cost margins 4326 6.2. Identifying and interpreting price–cost margins 4329 6.3. Summary 4333 7. Models of differentiated product competition 4334 7.1. Neoclassical demand models 4334